This demonstrates the Padua transform and inverse transform, explaining precisely the normalization and points

using FastTransforms

We define the Padua points and extract Cartesian components:

N = 15
x = pts[:,1]
y = pts[:,2];
nothing #hide

We take the Padua transform of the function:

f = (x,y) -> exp(x + cos(y))
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and use the coefficients to create an approximation to the function $f$:

f̃ = (x,y) -> begin
j = 1
ret = 0.0
for n in 0:N, k in 0:n
ret += f̌[j]*cos((n-k)*acos(x)) * cos(k*acos(y))
j += 1
end
ret
end
#3 (generic function with 1 method)

At a particular point, is the function well-approximated?

f̃(0.1,0.2) ≈ f(0.1,0.2)
true

Does the inverse transform bring us back to the grid?

ipaduatransform(f̌) ≈ f̃.(x,y)
true